for an answer specific to a personal situation, see your priest
in general until the first marriage is declared null by the Church the person is not free to date or become romantically involved with someone else, much less begin marriage preparations

If the person is Catholic and married outside Church law, they still need a declaration of invalidity due to lack of form first.

Not just from the whole point of adultery and sin - but unless one has made a break and healed from the past relationship - one is not ready for a new relationship - period end of story. Take it easy. Stop. Let Christ speak to your heart. Spend sometime alone with the blessed Sacrament and make sure you are even free to date first - go through the anullment process and be free - until you know what it is to be free and single - how can you really be sure this is the right thing?

I would be *very *concerned about someone who is already planning a marriage while still married to someone else, which would be the case if you are going through a divorce but it is not yet finalized. It doesn’t matter whether the marriage was civil or religious - you are married right now, so whoever your new fiance is came on the scene in an adulterous way. That does not speak well of the moral character of your fiance - are you sure he/she is someone you would really want to marry, assuming you even can? I am no canon lawyer, but I have heard that relationships that began in adultery cannot lead to valid marriages. You need to talk to your priest about this situation ASAP.

If someone is not married in the church but only civilly and is now going through a divorce, can that person start taking pre cana classes with someone else while still going through the divorce?

Based on my experience when getting married, the couple must meet first with the priest for the priest to even determine if the couple should even consider getting married before the couple even considers pre-cana classes. We met several times with our priest, together and individually before pre-cana classes were even consider. IMO, I can’t imagine any priest allowing a couple to even consider marriage if one of them is in the midst of a divorce let alone approving pre-cana classes. There is probably some liberal priest out there that would allow the couple to proceed, but it certainly sounds like a recipe for disaster.